1 hidden layer with 1000 neurons vs. 10 hidden layers with 100 neurons
Basically, having multiple layers (aka a deep network) makes your network more eager to recognize certain aspects of input data. For example, if you have the details of a house (size, lawn size, location etc.) as input and want to predict the price. The first layer may predict:


Big area, higher price
Small amount of bedrooms, lower price


The second layer might conclude:


Big area + small amount of bedrooms = large bedrooms = +- effect


Yes, one layer can also 'detect' the stats, however it will require more neurons as it cannot rely on other neurons to do 'parts' of the total calculation required to detect that stat.

Check out this answer